According to the Daily Mail, Parish was left confused by Sage's call to leave Canvot out of the matchday squad entirely for Saturday's 3-2 defeat to promoted Ipswich Town, a decision the Frenchman described as purely footballing.
Sage, appointed by Parish in the summer following Oliver Glasner's departure, said he had based the call on Canvot's performance against Fulham and his level in training that week.
"It was a sporting decision," Sage said. "I have to be honest with you, I didn't pick him in the team because of his game against Fulham and what he did during the week.
"I spoke with him. I need more from him. I'm sure he's about to do it and I wait for a good reaction on his part."
Canvot has emerged as a key part of the Crystal Palace project after a strong first season at Selhurst Park, establishing himself as a regular following Marc Guéhi's departure. Sage insisted his reasoning was long-term rather than an attempt to assert authority.
"I don't care about my authority, I just want the players at 100 per cent of what they are able to do, and when they give less, I have to shake them," he said. "My view is not short term, it's long term."
Canvot is not the only high-profile player to be left out under Sage this season, with Evann Guessand and Dwight McNeil also dropped for recent matches after what the head coach felt were below-par performances. Palace have lost three of their opening four league games and sit 16th in the table, with the club having already conceded 11 goals in five matches across all competitions this term.
Palace's defence was further stretched at Ipswich, where Axel Disasi was sent off, leaving Sage's side to play the majority of the match with 10 men before Ipswich substitute Zian Flemming struck late to seal the win.
Any tension between manager and chairman is unlikely to change Sage's approach to team selection, but an early frosty patch in the relationship is rarely a good sign for a new head coach still finding his feet.